Abstract
A set of linear second-order differential equations is converted into a semigroup, whose algebraic structure is used to generate novel equations. The Lagrangian formalism based on standard, null, and nonstandard Lagrangians is established for all members of the semigroup. For the null Lagrangians, their corresponding gauge functions are derived. The obtained Lagrangians are either new or generalization of those previously known. The previously developed Lie group approach to derive some equations of the semigroup is also described. It is shown that certain equations of the semigroup cannot be factorized, and therefore, their Lie groups cannot be determined. A possible solution of this problem is proposed, and the relationship between the Lagrangian formalism and the Lie group approach is discussed.
Highlights
Let Q be a set of all linear second-order differential equations (ODEs) of the form DyðxÞ = 0, where D ≡ d2/dx2 + BðxÞd/d x + CðxÞ, and BðxÞ and CðxÞ are smooth (C∞) and ordinary (B : R → R and C : R → R) functions to be determined
The Lagrangian formalism was established for all ODEs of the semigroup
We showed that the minimal Lagrangian is the simplest and the most fundamental, and that the other Lagrangians differ by the so-called null Lagrangians from the minimal one
Summary
Let us point out that an identity element may be formally introduced into the semigroup S by taking BðxÞ = CðxÞ = 0 and finding the solution yðxÞ = a0x + b0 with a0 and b0 being constants Having defined this identity element, the semigroup S becomes a monoid M, and since the original S is commutative, M is commutative. The Lagrangian formalism and the Lie group approach, the two independent methods of obtaining the ODEs of S, are compared. The advantages and disadvantages of each method are discussed, and it is shown that the Lagrangian formalism can be established for all considered ODEs; the Lie group approach is only limited to some ODEs that form a subsemigroup of S.
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